TranscriptSpecial Triangles

Grandpa Lindbergh looks at old photos and thinks about the past… Back in the day, he was a famous aviator. He sees a picture of an old friend. Oh boy, did they have some exciting times together. Grandpa wonders how his friend is doing now.
Suddenly, sick and tired of just sitting in his rocking chair and thinking back to his glory days, Grandpa decides to go on an expedition. He can visit an old friend. He packs up his things and gets his favorite girl ready to go – a double-decker named Kittyhawk. Off he goes into the wild, blue yonder.

Special Triangles

So far, the flight seems to be going well. Grandpa checks the plane’s flying angle: 45° and a distance of 2,450 feet. It’s all A-Okay. Wait. What’s that noise? Something is wrong. The engine just died, and he's falling at a 60° angle to the horizon! Oh no! Now the instrument panel is dead! Grandpa will have to crash land the plane.
How can we figure out Grandpa's location to rescue him? To figure this out, we can use special triangles. Let’s review what we know. Before the engine conked out, he flew at an angle of 45° and traveled 2,450 feet. After the engine blew, he fell at a 60° angle to the horizon. Let’s graph this. The angle next to the 60° falling angle equals 30°. As you can see, we divided the large triangle into two right triangles. The two right triangles are special triangles.

Iscoceles Triangles

First, let’s analyze the special triangle on the left. This left triangle has two 45° angles, so it's an iscoceles right triangle. How do we know this? There are 180° in a triangle, so do the math. If both angles have the same measurement then the legs opposite the angles also have the same measurement or length. Let’s add some labels, length 'a' goes here and here, and for the hypotenuse, we'll label it length 'c'.
We can use the Pythagorean Theorem to solve for the unknown measurement. For this special triangle, we get the equation c² = a² + a². By simplifying the right side of the equation and taking the square root on both sides of the equation, we get 'c' equals the square root of 2a². To calculate a square root of a product you can also calculate the product of the square root of each factor and simplify as far as possible. The hypotenuse of this special triangle is equal to 'a' times the square root of 2.
Now, let's apply this knowledge to Grandpa's problem. Since we know 'a' times the square root of 2 is equal to 2,450, we can solve for the variable 'a'. To make things easier, we'll round all numbers to the nearest whole number. The two equal legs have a length of 1,732 feet.

Equilateral Triangles

Okay, that's one down. Now, let’s work on the special triangle on the right. This is a 30-60-90 triangle. To help you to understand 30-60-90 triangles, lets draw an equilateral triangle. We'll label the length of each equal side as 2a. Now, as you can see the altitude splits the equilateral triangle into two 30-60-90 triangles. The base of each 30-60-90 triangle is equal to 'a'.
Now, let's apply this knowledge of 30-60-90 triangles to solve Lindbergh’s problem. We know the hypotenuse has a length of 2a, and the leg opposite the 30-degree angle has a length of 'a'. Let's call the third leg 'b'. We can use the Pythagorean Theorem to determine the length of the third leg with regard to 'a'. We get the equation (2a)² = a² + b².Simplifying the left side of the equation we get 4a² = a² + b². Now we can subtract a² on both sides of the equation, what leaves 3a² = b². Last taking the square root on both sides we get the root of (3a²) = b². Again, we can split up the left side, calculating the product of the square root of each factor. we get the root of 3 times the root of a² =b , so the leg opposite the 60° angle is equal to 'a' times the square root of 3, and since we already know that leg is equal to 1732 feet, we can solve for 'a' and calculate the other lengths. Remember to round to the nearest whole number. The hypotenuse of the 30-60-90 triangle is equal to 2,000 feet, and Grandpa Lindbergh traveled a total distance of 2,732 feet.

With no instruments, he managed the crash landing. That old guy is really something else! Is that Grandpa's friend!!?

About this Video Lesson

Description

Two special right triangles are the 30-60-90 and the 45-45-90 triangles.

For a 30-60-90 triangle, the ratios of the side lengths are 1:√3:2. The smallest side is opposite 30º while the longest side, or hypotenuse, is opposite 90º. In a 45-45-90 triangle, the ratio of the side lengths are 1:1:√2. The longest side, or hypotenuse, is opposite 90º. When solving for the missing length of a side of a special right triangle given the length of one of its sides, use the Pythagorean Theorem and take note of the relevant side ratio to use.

A knowledge of special triangles has several applications in engineering works and construction.